We introduce a local measure of the quality of a trial wave function: the local variance. Using this tool we examine the pair function often adopted to construct wave functions for small helium clusters and apply it to a wave function for 4 He 2 using the TTY interaction potential. Our analysis shows that this commonly employed functional form should be improved in the short-range region. We introduce a new model based on the short-range behaviour of the exact ground-state solution of a Morse potential. The resulting compact trial wave function for 4 He 2 recovers 98% of the exact binding energy with only six variational parameters.
We introduce a local measure of the quality of a trial wave function: the local variance. Using this tool we examine the pair function often adopted to construct wave functions for small helium clusters and apply it to a wave function for 4 He 2 using the TTY interaction potential. Our analysis shows that this commonly employed functional form should be improved in the short-range region. We introduce a new model based on the short-range behaviour of the exact ground-state solution of a Morse potential. The resulting compact trial wave function for 4 He 2 recovers 98% of the exact binding energy with only six variational parameters.
We calculated the 4He trimer and tetramer ground and excited states with the LM2M2 potential using our Gaussian expansion method for ab initio variational calculations of few-body systems. The method has been extensively used for a variety of three-, four-, and five-body systems in nuclear physics and exotic atomic and molecular physics. The trimer (tetramer) wave function is expanded in terms of symmetric three- (four-) body Gaussian basis functions, ranging from very compact to very diffuse, without assumption of any pair correlation function. The calculated results for the trimer ground and excited states are in excellent agreement with values reported in the literature. The binding energies of the tetramer ground and excited states are obtained as 558.98 and 127.33 mK (0.93 mK below the trimer ground state), respectively. We found that precisely the same shape of the short-range correlation (${r}_{ij}\ensuremath{\lesssim}4$ \AA{}) in the dimer appears in the ground and excited states of the trimer and tetramer. The overlap function between the trimer excited state and the dimer ground state and that between the tetramer excited state and the trimer ground state are almost proportional to the dimer wave function in the asymptotic region (up to $\ensuremath{\sim}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of 4HeN ($N\ensuremath{\ge}3$) measured from the 4He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where B2 is the dimer binding energy.
We report accurate ground state energies and structural properties for small clusters of 4He computed with the diffusion quantum Monte Carlo (DMC) method combined with high quality trial wave functions and using the recent analytical pair potential of Tang, Toennies, and Yiu [Phys. Rev. Lett. 74, 1546 (1995)]. Calculations based on the older HFD-B(He) potential are reported for comparison. The clusters are found to be extremely floppy and to be characterized by very diffuse wave functions. The DMC results for 4He2 and 4He3 are in excellent agreement with other calculations using conventional methods. 4He3 is found to have a noticeable contribution from nearly linear geometries. The internal structure of the clusters is described by a three particle correlation function which reveals a significantly non-spherical internal cluster structure. The energies for all cluster sizes are found to be slightly higher than those obtained with the HFD-B(He) pair potential. Exploratory calculations on the helium trimer indicate that the effects of three body interactions do not exceed the uncertainty margin provided by different state of the art pair potentials.
Abstract This paper describes improved trial wavefunctions for helium clusters and their use in variational QMC calculations of their binding energies, structural properties, and breathing mode vibrational excited states, for clusters of 3 to 270 atoms. The wavefunctions developed were similar to those of earlier work but included convenient analytic functions for two- and three-body correlations. All terms were of the Jastrow exponential form, based on sums of one-, two-, and three-body terms t1(ri), t2(rij), and t3(rij, rjk,rki), expressed as functions of interparticle distances. Excited-state wavefunctions for the breathing vibrational mode were obtained with the aid of procedures adapted from treatment of the quantum mechanics of liquid drop models. The interactions of helium atoms were specified as pairwise-additive with an accurate semi-empirical pair potential. Optimizations of parameters in the trial wavefunctions were carried out for individual cluster sizes with an exhaustive search procedure in a series of trial variational calculations, in which the parameters were varied to minimize the energy. A partial correlation of results was obtained by use of identical initial coordinates and random number seeds in these optimization calculations. Once the wavefunctions were optimized, longer calculations were used to determine properties of the clusters. In addition, diffusion QMC calculations were used to obtain more accurate nonvariational energies for comparisons.
Small helium (4He) clusters containing the lighter isotope He3 are studied by means of quantum Monte Carlo methods. Accurate ground state energies and structural properties are obtained using accurate trial wave functions and the Tang–Tonnies–Yiu (TTY) helium–helium pair potential. The dimer He4–3He is not bound; as well as the trimer He4He23. The smallest cluster containing He3 is He24He3 with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the He34 trimer, is able to bind another He3 atom, forming the tetramer He24He23, which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single He4 atom in a pure cluster with a He3 atom leads to an energetic destabilization, as the pair He4–3He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining He4 atoms, which retain the high floppiness already found in the pure clusters. As the number of atoms increases the isotopic impurity has the marked tendency to stay on the surface of the cluster. This behavior is consistent with the formation of the so-called “Andreev states” of a single He3 in liquid He4 helium and droplets, where the impurity tends to form single-particle states on the surface of the pure He4.
Abstract Small helium clusters Hen (n = 2, 3, ... , 10) in molecular beams produced by supersonic expansion of gaseous helium have been observed in several different experiments. Among these clusters the dimer, which has a single very slightly bound state, and the trimer, which could have many bound states, have especially unusual properties which QMC calculations can explain in detail. In this paper, diffusion QMC calculations were used to determine ground state properties for clusters (n = 2, ... , 10) using two accurate, but slightly different pairwise-additive potential energy expressions, a perturbation-based potential by Tang et al.a and an empirical potential (HFD-B) proposed by Aziz et al.b Trial functions for importance sampling were based on those determined earlier by Rick et alc. Expectation values and probability distributions were determined by the descendent-weighting technique to give results independent of the trial functions. As expected, a single bound state of the dimer was found, and its average internuclear distance was predicted to be 51.5 A. Energies for clusters with n = 2, 7 with the HFD-B potential were in good agreement with those of earlier calculations. Comparisons of the results for the two different potential energy expressions showed energies differing by a few percent. Tests of possible nonadditive interactions showed these to be far less important in their effects on predicted energies.
A trial wave function describing the ground state of a quantum system of N interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.